Metode Numerik 2 - Universitas Indonesia
Metode Numerik 2 - Universitas Indonesia
Metode Numerik 2 - Universitas Indonesia
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Dicari<br />
distribusi<br />
spasial .<br />
Persamaan Differensial Eliptik<br />
Bentuk umum PD eliptik: )<br />
r<br />
(<br />
ρ<br />
4π<br />
)<br />
r<br />
ψ(<br />
2<br />
<br />
<br />
−<br />
=<br />
∇<br />
Untuk kasus 2 dimensi: y)<br />
(x,<br />
ρ<br />
4π<br />
y)<br />
ψ(x,<br />
y<br />
x<br />
2<br />
2<br />
2<br />
2<br />
−<br />
=<br />
⎟<br />
⎟<br />
⎠<br />
⎞<br />
⎜<br />
⎜<br />
⎝<br />
⎛<br />
∂<br />
∂<br />
+<br />
∂<br />
∂<br />
Gunakan metode finite differences:<br />
)<br />
y<br />
y<br />
x<br />
x<br />
(h<br />
h<br />
ψ<br />
2ψ<br />
ψ<br />
h<br />
)<br />
y<br />
,<br />
ψ(x<br />
)<br />
y<br />
,<br />
(x<br />
2ψ<br />
)<br />
y<br />
,<br />
ψ(x<br />
y)<br />
ψ(x,<br />
y<br />
h<br />
ψ<br />
2ψ<br />
ψ<br />
h<br />
)<br />
y<br />
,<br />
ψ(x<br />
)<br />
y<br />
,<br />
(x<br />
2ψ<br />
)<br />
y<br />
,<br />
ψ(x<br />
y)<br />
ψ(x,<br />
x<br />
j<br />
1<br />
j<br />
i<br />
1<br />
i<br />
2<br />
1<br />
-<br />
j<br />
i,<br />
j<br />
i,<br />
1<br />
j<br />
i,<br />
2<br />
1<br />
-<br />
j<br />
i<br />
j<br />
i<br />
1<br />
j<br />
i<br />
y<br />
,<br />
x<br />
2<br />
2<br />
2<br />
j<br />
1,<br />
-<br />
i<br />
j<br />
i,<br />
j<br />
1,<br />
i<br />
2<br />
j<br />
1<br />
-<br />
i<br />
j<br />
i<br />
j<br />
1<br />
i<br />
y<br />
,<br />
x<br />
2<br />
2<br />
j<br />
i<br />
j<br />
i<br />
−<br />
=<br />
−<br />
=<br />
+<br />
−<br />
=<br />
+<br />
−<br />
≈<br />
∂<br />
∂<br />
+<br />
−<br />
=<br />
+<br />
−<br />
≈<br />
∂<br />
∂<br />
+<br />
+<br />
+<br />
+<br />
+<br />
+<br />
Diperoleh: [ ]<br />
1<br />
-<br />
j<br />
i,<br />
1<br />
j<br />
i,<br />
j<br />
1,<br />
-<br />
i<br />
j<br />
1,<br />
i<br />
4<br />
1<br />
j<br />
i,<br />
2<br />
j<br />
i,<br />
ψ<br />
ψ<br />
ψ<br />
ψ<br />
ρ<br />
π<br />
h<br />
ψ +<br />
+<br />
+<br />
+<br />
= +<br />
+<br />
ψ<br />
126